The quantum mechanics of two dimensional superfluids Physical Review B 71, 144508 and 144509 (2005), cond-mat/0502002 Leon Balents (UCSB) Lorenz Bartosch (Yale) Anton Burkov (UCSB) Subir Sachdev (Yale) Krishnendu Sengupta (Toronto) Talk online: Sachdev OutlineOutline I. Bose-Einstein condensation and superfluidity II. The superfluid-Mott insulator quantum phase transition III. The cuprate superconductors Superfluids proximate to finite doping Mott insulators with VBS order ? IV. Vortices in the superfluid V. Vortices in superfluids near the superfluid-insulator quantum phase transition The “quantum order” of the superconducting state: evidence for vortex flavors I. Bose-Einstein condensation and superfluidity Superfluidity/superconductivity occur in: • liquid 4He • metals Hg, Al, Pb, Nb, Nb3Sn….. • liquid 3He • neutron stars • cuprates La2-xSrxCuO4, YBa2Cu3O6+y…. •M3C60 • ultracold trapped atoms •MgB2 The Bose-Einstein condensate: A macroscopic number of bosons occupy the lowest energy quantum state Such a condensate also forms in systems of fermions, where the bosons are Cooper pairs of fermions: ky Pair wavefunction in cuprates: Ψ =−kk22↑↓−↓↑ k ()xy() x S = 0 Velocity distribution function of ultracold 87Rb atoms M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269, 198 (1995) Superflow: The wavefunction of the condensate iθ r Ψ→Ψe ( ) Superfluid velocity v = ∇θ s m (for non-Galilean invariant superfluids, the co-efficient of ∇θ is modified) II. The superfluid-Mott insulator quantum phase transition Apply a periodic potential (standing laser beams) to trapped ultracold bosons (87Rb) Momentum distribution function of bosons Bragg reflections of condensate at reciprocal lattice vectors M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Momentum distribution function of bosons Bragg reflections of condensate at reciprocal lattice vectors M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Superfluid-insulator quantum phase transition at T=0 V0=0Er V0=3Er V0=7Er V0=10Er V0=13Er V0=14Er V0=16Er V0=20Er Superfluid-insulator quantum phase transition at T=0 M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Bosons at filling fraction f = 1 Weak interactions: superfluidity Strong interactions: Mott insulator which preserves all lattice symmetries M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Bosons at filling fraction f = 1 Ψ ≠ 0 Weak interactions: superfluidity Bosons at filling fraction f = 1 Ψ ≠ 0 Weak interactions: superfluidity Bosons at filling fraction f = 1 Ψ ≠ 0 Weak interactions: superfluidity Bosons at filling fraction f = 1 Ψ ≠ 0 Weak interactions: superfluidity Bosons at filling fraction f = 1 Ψ = 0 Strong interactions: insulator Bosons at filling fraction f = 1/2 Ψ ≠ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Bosons at filling fraction f = 1/2 Ψ ≠ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Bosons at filling fraction f = 1/2 Ψ ≠ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Bosons at filling fraction f = 1/2 Ψ ≠ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Bosons at filling fraction f = 1/2 Ψ ≠ 0 Weak interactions: superfluidity C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Bosons at filling fraction f = 1/2 Ψ = 0 Strong interactions: insulator C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Bosons at filling fraction f = 1/2 Ψ = 0 Strong interactions: insulator C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Insulating phases of bosons at filling fraction f = 1/2 1 = ( + ) 2 Charge density Valence bond Valence bond wave (CDW) order solid (VBS) order solid (VBS) order iQr. Can define a common CDW/VBS order using a generalized "density" ρρ(r ) = ∑ Qe Q All insulators have Ψ=0 and ρQ ≠0 for certain Q C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002) Superfluid-insulator transition of bosons at generic filling fraction f The transition is characterized by multiple distinct order parameters (boson condensate, VBS/CDW order) Traditional (Landau-Ginzburg-Wilson) view: Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid.